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So, generally, QuEST uses Newton Raphson method starting with [[File:Equation44.png]] calculated above, and over multiple iterations reaches at a more accurate value for λ.<br \>
Notice how this is different from the qmethod. QuEST doesn’t directly solve for the eigen values. It starts with a reasonable guess and then moves on to check out if there are values close to this guess that can do better.<br \>
Now to solve for the quaternion, both scalar and vector parts are divided <math>q_4 =cos \frac{\phi}{2}</math> (<math>q_4</math> is the scalar part; <math> \phi</math> is the angle of rotation). Note, this only changes the magnitude of the quaternion and the final answer can be corrected for later by normalising it. So, the vector part now is <math>\boldsymbol{p} = \frac{\bar{\boldsymbol{q}}}{q_4} = atana*tan\frac{\phi}{2}</math>,
where <math>a</math> is the vector along axis of rotation. The initial matrix equation now turns out as follows
[[File:Equation48.png|frame|center]]